Mathematics > Dynamical Systems

Title:Outer Billiards, Arithmetic Graphs, and the Octagon

Abstract: Outer Billiards is a geometrically inspired dynamical system based on a
convex shape in the plane.
When the shape is a polygon, the system has a combinatorial flavor. In the
polygonal case, there is a natural acceleration of the map, a first return map
to a certain strip in the plane. The arithmetic graph is a geometric encoding
of the symbolic dynamics of this first return map.
In the case of the regular octagon, the case we study, the arithmetic graphs
associated to periodic orbits are polygonal paths in R^8. We are interested in
the asymptotic shapes of these polygonal paths, as the period tends to
infinity. We show that the rescaled limit of essentially any sequence of these
graphs converges to a fractal curve that simultaneously projects one way onto a
variant of the Koch snowflake and another way onto a variant of the Sierpinski
carpet. In a sense, this gives a complete description of the asymptotic
behavior of the symbolic dynamics of the first return map.
What makes all our proofs work is an efficient (and basically well known)
renormalization scheme for the dynamics.

Comments:

86 pages, mildly computer-aided proof. My java program this http URL illustrates essentially all the ideas in the paper in an interactive and well-documented way. This is the second version. The only difference from the first version is that I simplified the proof of Main Theorem, Statement 2, at the end of Ch. 8